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A unit vector of modulus 2 is equally...

A unit vector of modulus 2 is equally inclined to `x` - and `y` -axes angle at an angle `pi//3` . Find the length of projection of the vector on the `z` -axis.

Text Solution

Verified by Experts

Given that the vector is inclined at an angle `pi//3` with both x- and y-axes. Then
`" "cosalpha=cosbeta=(1)/(2)`
Also we know that `cos^(2)alpha +cos^(2)beta+cos^(2)gamma=1`. Therefore,
`" "cos^(2)gamma=(1)/(2)`
or `" "cosgamma=pm(1)/(sqrt(2))`
Thus, the given vector is
`" "2(cosalphahati+cosbetahatj+cosgammahatk)=2((hati)/(2)+(hatj)/(2)pm(hatk)/(sqrt(2)))=hati+hatjpmsqrt(2)hatk`
Hence, the length of projection of vector on the z-axis is `sqrt(2)` units.
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